Complete the Square

Pronunciation: /kəmˈplit ðə skwɛər/ ?

Complete the square is an algorithm used to convert a quadratic equation into vertex form.[1] The complete the square algorithm is also used to derive the quadratic equation. Table 1 gives the algorithm for completing the square.

StepQuadratic EquationVertex FormDescription
1Start with the equation in standard form.
2Use the distributive property of multiplication of addition and subtraction to get x2+bx inside parenthesis.
3Simplify, if possible.
4Add and subtract b2/(4a2).
5Simplify, if possible.
6Use the distributive property of multiplication over addition and subtraction to move -(b/2a)2 out of the parenthesis.
7Simplify if possible.
8Factor the square.
9Subtract the constant term from both sides. The equation is now in vertex form. This demonstration will continue to derive the quadratic formula by solving for x.
10Empty SpaceDivide both sides by a.
11Empty SpaceMake 4a2 the common denominator.
12Empty SpaceCombine the fractions.
13Empty SpaceTake the square root of both sides.
14Empty SpaceSimplify.
15Empty SpaceSubtract b/(2a) from both sides.
16Empty SpaceCombine the fractions.
17Empty SpaceUse the symmetric property of equality to put the x on the left of the equals sign. This is the quadratic formula.
Table 1: The complete the square algorithm

Geometric Interpretation of Complete the Square

1The square has dimensions x by x so the square represents x2. The rectangle has dimensions b by x, so the rectangle represents bx. The circle represents a.
2The rectangle can be divided in half. Each half represents bx/2.
3empty space
4Move the two rectangles around the square to form part of a larger square.
5Complete the square by adding b2/4 to both sides.
Table 2: Geometric interpretation of complete the square.


  1. Wells, Webster. A Complete Course is Algebra for Academies and High Schools, pp 213-217. Wells' Mathematical Series. Leach, Shewell & Sandborn, 1885. (Accessed: 2010-01-16).
  2. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry, pp 206-209. International Textbook Company, January 1963. (Accessed: 2010-01-17).

More Information

  • McAdams, David. Vertex Form. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-12-13.
  • Stapel, Elizabeth. Completing the Square: Finding the Vertex. PurpleMath. 2009-12-13.

Cite this article as:

Complete the Square. 2010-01-17. All Math Words Encyclopedia. Life is a Story Problem LLC.


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Revision History

2010-01-17: Added "References" (McAdams, David.)
2008-11-25: Changed equations to images (McAdams, David.)
2008-10-17: Initial version (McAdams, David.)

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